Ali Enayat, University of Gothenburg
PA with a class of indiscernibles

This talk focuses on the theory PAI (I for Indiscernibles), a theory formulated in the language of PA augmented with a unary predicate I(x). Models of PAI are of the form (M,I) where (1) M is a model of PA, (2) I is a proper class of M, i.e., I is unbounded in M and (M,I) satisfies PA*, and (3) I forms a class of indiscernibles over M. The formalizability of the Infinite Ramsey Theorem in PA makes it clear that PAI is a conservative extension of PA. As we will see, nonstandard models of PA (of any cardinality) that have an expansion to a model of PAI are precisely those nonstandard models PA that can carry an inductive partial satisfaction class. The formulation and investigation of PAI was inspired by my work on the set theoretical sibling ZFI of PAI, whose behavior I will also compare and contrast with that of PAI.

Speaker:     Joshua Sussan, Medgar Evers, CUNY.

Title:    Categorification and quantum topology.

Abstract: The Jones polynomial of a link could be defined through the representation theory of quantum sl(2). It leads to a 3-manifold invariant and 2+1 dimensional TQFT. In the mid 1990s, Crane and Frenkel outlined the categorification program with the aim of constructing a 3+1 dimensional TQFT by upgrading the representation theory of quantum sl(2) to some categorical structures. We will review these ideas and give examples of various categorifications of quantum sl(2) constructions.

Hossein Lamei Ramandi, Cornell University
Galvin's question on non-σσ-well ordered linear orders

Assume CC is the class of all linear orders LL such that LL is not a countable union of well ordered sets, and every uncountable subset of LL contains a copy of ω1ω1. We show it is consistent that CC has minimal elements. This answers an old question due to Galvin.

Title: Substructural Solutions to the Semantic Paradoxes: a Dialetheic Perspective

Abstract: Over the last decade or so, a number of writers have argued for  solutions to the paradoxes of semantic self-reference which proceed by dropping some of the structural rules of inference, most notably Cut and/or Contraction.  In this paper, we will examine such accounts, with a particular eye on their relationship to more familiar dialetheic accounts.

Independence in PA: The Method of (L,n)(L,n)-Models

The purpose of this talk is to exposit a method for proving independence over PA of 'mathematical' statements (whatever that means). The method uses the concept of an (L,n)(L,n)-model: a finite sequence of finite LL-structures for some first order LL extending the language of arithmetic. The idea is that this finite sequence is intended to represent increasing approximations of a potentially infinite structure and the machinery developed allows one to translate (meta-mathematical) compactness type statements, which are easily seen to be independent of PA, into statements about finite combinatorics, which have 'mathematical content'. (L,n)(L,n)-models were introduced by Shelah in the 70's in his alternative proof of the Paris-Harrington Theorem and also appears (implicitly) in his example of a true, unprovable Π01Π10 statement of some 'mathematical' content. A similar idea was discovered independently by Kripke (unpublished). In this talk we will flesh out the details of this method and extend the general theory. This will allow us to present, in a fairly systematic fashion, proofs of the Paris-Harrington Theorem and the independence over PA of several, similar, Ramsey Theoretic statements including some which are Π01Π10.

Farmer Schlutzenberg, University of Münster
(Non)uniqueness and (un)definability of embeddings beyond choice

Work in ZF and let j:Vα→Vαj:Vα→Vα be an elementary, or partially elementary, embedding. One can examine the degree of uniqueness, definability or constructibility of jj. For example, is there β<αβ<α such that jj is the unique (partially) elementary extension of j↾Vβj↾Vβ? Is jj definable from parameters over VαVα? We will discuss some results along these lines, illustrating that answers can depend heavily on circumstances. Some of the work is due independently and earlier to Gabriel Goldberg.

Hiroshi Sakai, Kobe University
Generalized stationary reflection and cardinal arithmetic

The stationary reflection principle, which is often called the Weak Reflection Principle, is known to have many interesting consequences. As for cardinal arithmetic, it implies that λω=λλω=λ for all regular cardinal λ≥ω2λ≥ω2. In this talk, we will discuss higher analogues of this stationary reflection principle and their consequences on cardinal arithmetic.

Mateusz Łełyk, University of Warsaw
Nonequivalent axiomatizations of PAPA and the Tarski Boundary

We study a family of axioms expressing‘All axioms of PA are true.' (*)‘All axioms of PA are true.' (*)where PA denotes Peano Arithmetic. More precisely, each such axiom states that all axioms from a chosen axiomatization of PA are true. We start with a very natural theory of truth CT−(PA)CT−(PA) which is a finite extension of PA in the language of arithmetic augmented with a fresh predicate T to serve as a truth predicate for the language of arithmetic. Additional axioms of this theory are straightforward translations of inductive Tarski truth conditions. To study various possible ways of expressing (*), we investigate extensions of CT−(PA)CT−(PA) with axioms of the form∀x(δ(x)→T(x)).∀x(δ(x)→T(x)).In the above (and throughout the whole abstract) δ(x)δ(x) is an elementary formula which is proof-theoretically equivalent to the standard axiomatization of PA with the induction scheme, i.e. the equivalence∀x(Provδ(x)≡ProvPA(x)).∀x(Provδ(x)≡ProvPA(x)).is provable in IΣ1IΣ1. For every such δδ, the extension of CT−(PA)CT−(PA) with the above axiom will be denoted CT−[δ]CT−[δ].

In particular we shall focus on the arithmetical strength of theories CT−[δ]CT−[δ]. The 'line' demarcating extensions of CT−(PA)CT−(PA) which are conservative over PA from the nonconservative ones is known in the literature as the Tarski Boundary. For some time, there seemed to be the least (in terms of deductive strength) *natural* extension of CT−(PA)CT−(PA) on the nonconservative side of the boundary, whose one axiomatization is given by CT−(PA)CT−(PA) and Δ0Δ0 induction for the extended language (the theory is called CT0CT0). This theory can equivalently be axiomatized by adding to CT−(PA)CT−(PA) the natural formal representation of the statement 'All theorems of PAPA are true.'. We show that the situation between the Tarski Boundary and CT0CT0 is much more interesting:

Theorem 1: For every r.e. theory Th in the language of arithmetic the following are equivalent:
1) CT0⊢CT0⊢ Th
2) there exists δδ such that CT−[δ]CT−[δ] and Th have the same arithmetical consequences.

Theorem 1 can be seen as a representation theorem for r.e. theories below REFω(PA)REFω(PA) (all finite iterations of uniform reflection over PAPA, which is the set of arithmetical consequences of CT0CT0): each such theory can be finitely axiomatized by a theory of the form CT−[δ]CT−[δ], where δδ is proof-theoretically reducible to PAPA.

Secondly, we use theories CT−[δ]CT−[δ] to investigate the situation below the Tarski Boundary. We shall prove the following result

Theorem 2: There exists a family {δf}f∈ω<ω{δf}f∈ω<ω such that for all f,g∈ω<ωf,g∈ω<ω
1) CT−[δf]CT−[δf] is conservative over PAPA;
2) if f⊊gf⊊g, then CT−[δg]CT−[δg] properly extends CT−[δf]CT−[δf];
3) if f⊥gf⊥g then CT−[δg]∪CT−[δf]CT−[δg]∪CT−[δf] is nonconservative over PAPA (but consistent).

Magidor-style embedding characterizations of large cardinals

Motivated by a classical theorem of Magidor, I will present results providing characterizations of important objects from the lower end of the large cardinal hierarchy through the existence of elementary embeddings between set-sized models that map their critical point to the large cardinal in question. Focusing on the characterization of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, I will show how these results can be used in the study of the combinatorics of strong chain conditions and the investigation of principles of structural reflection formulated by Bagaria.

Farmer Schlutzenberg, University of Münster
(Non)uniqueness and (un)definability of embeddings beyond choice

Work in ZF and let j:Vα→Vαj:Vα→Vα be an elementary, or partially elementary, embedding. One can examine the degree of uniqueness, definability or constructibility of jj. For example, is there β<αβ<α such that jj is the unique (partially) elementary extension of j↾Vβj↾Vβ? Is jj definable from parameters over VαVα? We will discuss some results along these lines, illustrating that answers can depend heavily on circumstances. Some of the work is due independently and earlier to Gabriel Goldberg.

Leszek Kołodziejczyk, University of Warsaw

An isomorphism theorem for models of Weak Kőnig's Lemma without induction

We prove that any two countable models of the theory WKL∗0WKL0∗ sharing the same first-order universe and containing the same counterexample to Σ01Σ10 induction are isomorphic.

This theorem implies that over WKL∗0+¬IΣ01WKL0∗+¬IΣ10, the analytic hierarchy collapses to the arithmetic hierarchy. It also implies that WKL∗0WKL0∗ is the strongest Π12Π21 statement that is Π11Π11-conservative over RCA∗0+¬IΣ01RCA0∗+¬IΣ10. Together with the (slightly subtle) generalizations of the theorem to higher levels of the arithmetic hierarchy, this gives an 'almost negative' answer to a question of Towsner, who asked whether Π11Π11-conservativity of Π12Π21 sentences over collection principles is a Π02Π20-complete computational problem. Our results also have some implications for the reverse mathematics of combinatorial principles: for instance, we get a specific Π11Π11 sentence that is provable in RCA0+BΣ02RCA0+BΣ20 exactly if the Π11Π11 consequences of RCA0+RT22RCA0+RT22 coincide with BΣ02BΣ20.

On the side, we also give a positive answer to Towsner's question as originally stated.

Joint work with Marta Fiori Carones, Tin Lok Wong, and Keita Yokoyama.

Mateusz Łełyk, University of Warsaw
Nonequivalent axiomatizations of PAPA and the Tarski Boundary

We study a family of axioms expressing‘All axioms of PA are true.' (*)‘All axioms of PA are true.' (*)where PA denotes Peano Arithmetic. More precisely, each such axiom states that all axioms from a chosen axiomatization of PA are true. We start with a very natural theory of truth CT−(PA)CT−(PA) which is a finite extension of PA in the language of arithmetic augmented with a fresh predicate T to serve as a truth predicate for the language of arithmetic. Additional axioms of this theory are straightforward translations of inductive Tarski truth conditions. To study various possible ways of expressing (*), we investigate extensions of CT−(PA)CT−(PA) with axioms of the form∀x(δ(x)→T(x)).∀x(δ(x)→T(x)).In the above (and throughout the whole abstract) δ(x)δ(x) is an elementary formula which is proof-theoretically equivalent to the standard axiomatization of PA with the induction scheme, i.e. the equivalence∀x(Provδ(x)≡ProvPA(x)).∀x(Provδ(x)≡ProvPA(x)).is provable in IΣ1IΣ1. For every such δδ, the extension of CT−(PA)CT−(PA) with the above axiom will be denoted CT−[δ]CT−[δ].

In particular we shall focus on the arithmetical strength of theories CT−[δ]CT−[δ]. The 'line' demarcating extensions of CT−(PA)CT−(PA) which are conservative over PA from the nonconservative ones is known in the literature as the Tarski Boundary. For some time, there seemed to be the least (in terms of deductive strength) *natural* extension of CT−(PA)CT−(PA) on the nonconservative side of the boundary, whose one axiomatization is given by CT−(PA)CT−(PA) and Δ0Δ0 induction for the extended language (the theory is called CT0CT0). This theory can equivalently be axiomatized by adding to CT−(PA)CT−(PA) the natural formal representation of the statement 'All theorems of PAPA are true.'. We show that the situation between the Tarski Boundary and CT0CT0 is much more interesting:

Theorem 1: For every r.e. theory Th in the language of arithmetic the following are equivalent:
1) CT0⊢CT0⊢ Th
2) there exists δδ such that CT−[δ]CT−[δ] and Th have the same arithmetical consequences.

Theorem 1 can be seen as a representation theorem for r.e. theories below REFω(PA)REFω(PA) (all finite iterations of uniform reflection over PAPA, which is the set of arithmetical consequences of CT0CT0): each such theory can be finitely axiomatized by a theory of the form CT−[δ]CT−[δ], where δδ is proof-theoretically reducible to PAPA.

Secondly, we use theories CT−[δ]CT−[δ] to investigate the situation below the Tarski Boundary. We shall prove the following result

Theorem 2: There exists a family {δf}f∈ω<ω{δf}f∈ω<ω such that for all f,g∈ω<ωf,g∈ω<ω
1) CT−[δf]CT−[δf] is conservative over PAPA;
2) if f⊊gf⊊g, then CT−[δg]CT−[δg] properly extends CT−[δf]CT−[δf];
3) if f⊥gf⊥g then CT−[δg]∪CT−[δf]CT−[δg]∪CT−[δf] is nonconservative over PAPA (but consistent).

James Walsh, Cornell University
Reducing omega-model reflection to iterated syntactic reflection

Two types of principles are commonly called “reflection principles” in reverse mathematics. According to syntactic reflection principles for T, every theorem of T (from some complexity class) is true. According to semantic reflection principles, every set belongs to some (sufficiently correct) model of T. We will present a connection between syntactic reflection and semantic reflection in second-order arithmetic: for any Pi^1_2 axiomatized theory T, every set is contained in an omega model of T if and only if every iteration of Pi^1_1 reflection for T along a well-ordering is Pi^1_1 sound. There is a thorough proof-theoretic understanding of the latter in terms of ordinal analysis. Accordingly, these reductions yield proof-theoretic analyses of omega-model reflection principles. This is joint work with Fedor Pakhomov.

Date and Time:     Wednesday February 3, 2021, 7:00 - 8:30 PM., on Zoom.

Title:    Isotropy Groups of Quasi-Equational Theories.

Abstract: In [2], my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in [1]) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.

[1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012.

[2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018.

Date and Time:     Wednesday February 10, 2021, 7:00 - 8:30 PM., on Zoom.

Title:    Invertibility in Operads : an elementary arithmetic approach.

Abstract: This talk is motivated by two areas of 'lost mathematics' -- topics where it is clear that interesting theory was once known & understood, but only incomplete traces remain in the historical record. One of these was due to ancient Greek mathematicians & logicians, and the other is a much lesser-known relation of a famous open problem from the 20th century.

One objective of this talk is to trace a link between the two. However, this is not an exercise in the 'History of Mathematics' -- the connections rely on theory that certainly was not understood in either time period.

Precisely, we consider 'Invertible Operads' -- that is, those whose composition operations are either partially or globally invertible. We look at examples that are freely generated by some given set of operations, with particular reference to those whose composition operations may be given by elementary arithmetic functions.

We demonstrate how such structures arise in a range of different topics, providing previously unobserved connections between them. This includes subjects such as standard Young tableaux, mixed-radix counting systems, topologies on the natural numbers, logical models, famous groups, and combinatorially-inspired polyhedra.

This is very much work in progress, and aims to present interesting questions as much as interesting structures and results.